linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp

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• Last update: 2023-12-26 21:26:22+09:00
• Problem: https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A

Depends on

• convolution/ntt.hpp
• 線形漸化式の発見・第 $N$ 項推定 (formal_power_series/linear_recurrence.hpp)
• Black box linear algebra を利用した各種高速計算 (linear_algebra_matrix/blackbox_algorithm.hpp)
• Black box linear algebra のための行列 (linear_algebra_matrix/blackbox_matrices.hpp)
• linear_algebra_matrix/matrix.hpp
• modint.hpp

Code

#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY
#include "../../modint.hpp"
#include "../blackbox_algorithm.hpp"
#include "../blackbox_matrices.hpp"
#include "../matrix.hpp"
#include <chrono>
#include <cstdio>
#include <random>
#include <type_traits>
#include <vector>
using namespace std;

mt19937 mt(1010101);

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, toeplitz_ntt<MODINT>>::value>::type * = nullptr>
toeplitz_ntt<MODINT> gen_random_matrix() {
    auto rnd = uniform_int_distribution<int>(1, 100);
    int h = rnd(mt), w = max(h, rnd(mt));
    auto vh = gen_random_vector<MODINT>(h), vw = gen_random_vector<MODINT>(w);
    vw[0] = vh[0];
    return toeplitz_ntt<MODINT>(vh, vw);
}

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, square_toeplitz_ntt<MODINT>>::value>::type * = nullptr>
square_toeplitz_ntt<MODINT> gen_random_matrix() {
    int N = uniform_int_distribution<int>(1, 100)(mt);
    auto v = gen_random_vector<MODINT>(N * 2 - 1);
    return square_toeplitz_ntt<MODINT>(v);
}

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, sparse_matrix<MODINT>>::value>::type * = nullptr>
sparse_matrix<MODINT> gen_random_matrix() {
    int H = uniform_int_distribution<int>(1, 20)(mt),
        W = max(H, uniform_int_distribution<int>(1, 20)(mt));
    sparse_matrix<MODINT> M(H, W);
    for (int i = 0; i < H; i++) {
        for (int j = 0; j < W; j++) {
            MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt);
            M.add_element(i, j, v);
        }
    }
    return M;
}

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, matrix<MODINT>>::value>::type * = nullptr>
matrix<MODINT> gen_random_matrix() {
    int H = uniform_int_distribution<int>(1, 20)(mt),
        W = max(H, uniform_int_distribution<int>(1, 20)(mt));
    matrix<MODINT> M(H, W);
    for (int i = 0; i < H; i++) {
        for (int j = 0; j < W; j++) {
            MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt);
            M[i][j] = v;
        }
    }
    return M;
}

template <typename MATRIX, typename mint> void blackbox_mat_checker(int n_try) {
    for (int i = 0; i < n_try; i++) {
        const MATRIX M = gen_random_matrix<mint, MATRIX>();
        vector<vector<mint>> Mvv = M.vecvec();
        const int H = M.height(), W = M.width();
        const auto b = gen_random_vector<mint>(H);
        // Check linear eq. solver
        const auto x = linear_system_solver_lanczos(M, b);
        assert(M.prod(x) == b);

        // Check prod()
        const auto c = gen_random_vector<mint>(W);
        const auto Mc = M.prod(c);
        vector<mint> Mc2(H);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) Mc2[i] += Mvv[i][j] * c[j];
        }
        assert(Mc == Mc2);

        // Check prod_left()
        const auto a = gen_random_vector<mint>(H);
        const auto aM = M.prod_left(a);
        vector<mint> aM2(W);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) aM2[j] += Mvv[i][j] * a[i];
        }
        assert(aM == aM2);

        // Check black_box_determinant()
        if (H == W) {
            mint det = blackbox_determinant<MATRIX, mint>(M);
            mint det2 = matrix<mint>(Mvv).gauss_jordan().determinant_of_upper_triangle();
            assert(det == det2);
        }
    }
}

int main() {
    using mint = ModInt<998244353>;
    blackbox_mat_checker<toeplitz_ntt<mint>, mint>(1000);
    blackbox_mat_checker<square_toeplitz_ntt<mint>, mint>(1000);
    blackbox_mat_checker<sparse_matrix<mint>, mint>(1000);
    blackbox_mat_checker<matrix<mint>, mint>(1000);
    puts("Hello World");
}

#line 1 "linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp"
#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY
#line 2 "modint.hpp"
#include <cassert>
#include <iostream>
#include <set>
#include <vector>

template <int md> struct ModInt {
    using lint = long long;
    constexpr static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val_;
    int val() const noexcept { return val_; }
    constexpr ModInt() : val_(0) {}
    constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
    constexpr ModInt(lint v) { _setval(v % md + md); }
    constexpr explicit operator bool() const { return val_ != 0; }
    constexpr ModInt operator+(const ModInt &x) const {
        return ModInt()._setval((lint)val_ + x.val_);
    }
    constexpr ModInt operator-(const ModInt &x) const {
        return ModInt()._setval((lint)val_ - x.val_ + md);
    }
    constexpr ModInt operator*(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.val_ % md);
    }
    constexpr ModInt operator/(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.inv().val() % md);
    }
    constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
    constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
    friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
    friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
    friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
    constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
    constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
    constexpr bool operator<(const ModInt &x) const {
        return val_ < x.val_;
    } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
        return os << x.val_;
    }

    constexpr ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }

    static constexpr int cache_limit = std::min(md, 1 << 21);
    static std::vector<ModInt> facs, facinvs, invs;

    constexpr static void _precalculation(int N) {
        const int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }

    constexpr ModInt inv() const {
        if (this->val_ < cache_limit) {
            if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
            while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val_];
        } else {
            return this->pow(md - 2);
        }
    }
    constexpr ModInt fac() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val_];
    }
    constexpr ModInt facinv() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val_];
    }
    constexpr ModInt doublefac() const {
        lint k = (this->val_ + 1) / 2;
        return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                                : ModInt(k).fac() * ModInt(2).pow(k);
    }

    constexpr ModInt nCr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
    }

    constexpr ModInt nPr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv();
    }

    static ModInt binom(int n, int r) {
        static long long bruteforce_times = 0;

        if (r < 0 or n < r) return ModInt(0);
        if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);

        r = std::min(r, n - r);

        ModInt ret = ModInt(r).facinv();
        for (int i = 0; i < r; ++i) ret *= n - i;
        bruteforce_times += r;

        return ret;
    }

    // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
    // Complexity: O(sum(ks))
    template <class Vec> static ModInt multinomial(const Vec &ks) {
        ModInt ret{1};
        int sum = 0;
        for (int k : ks) {
            assert(k >= 0);
            ret *= ModInt(k).facinv(), sum += k;
        }
        return ret * ModInt(sum).fac();
    }

    // Catalan number, C_n = binom(2n, n) / (n + 1)
    // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
    // https://oeis.org/A000108
    // Complexity: O(n)
    static ModInt catalan(int n) {
        if (n < 0) return ModInt(0);
        return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
    }

    ModInt sqrt() const {
        if (val_ == 0) return 0;
        if (md == 2) return val_;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val_, md - x.val_));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};

using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 2 "formal_power_series/linear_recurrence.hpp"
#include <algorithm>
#line 4 "formal_power_series/linear_recurrence.hpp"
#include <utility>
#line 6 "formal_power_series/linear_recurrence.hpp"

// CUT begin
// Berlekamp–Massey algorithm
// https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm
// Complexity: O(N^2)
// input: S = sequence from field K
// return: L          = degree of minimal polynomial,
//         C_reversed = monic min. polynomial (size = L + 1, reversed order, C_reversed[0] = 1))
// Formula: convolve(S, C_reversed)[i] = 0 for i >= L
// Example:
// - [1, 2, 4, 8, 16]   -> (1, [1, -2])
// - [1, 1, 2, 3, 5, 8] -> (2, [1, -1, -1])
// - [0, 0, 0, 0, 1]    -> (5, [1, 0, 0, 0, 0, 998244352]) (mod 998244353)
// - []                 -> (0, [1])
// - [0, 0, 0]          -> (0, [1])
// - [-2]               -> (1, [1, 2])
template <typename Tfield>
std::pair<int, std::vector<Tfield>> find_linear_recurrence(const std::vector<Tfield> &S) {
    int N = S.size();
    using poly = std::vector<Tfield>;
    poly C_reversed{1}, B{1};
    int L = 0, m = 1;
    Tfield b = 1;

    // adjust: C(x) <- C(x) - (d / b) x^m B(x)
    auto adjust = [](poly C, const poly &B, Tfield d, Tfield b, int m) -> poly {
        C.resize(std::max(C.size(), B.size() + m));
        Tfield a = d / b;
        for (unsigned i = 0; i < B.size(); i++) C[i + m] -= a * B[i];
        return C;
    };

    for (int n = 0; n < N; n++) {
        Tfield d = S[n];
        for (int i = 1; i <= L; i++) d += C_reversed[i] * S[n - i];

        if (d == 0)
            m++;
        else if (2 * L <= n) {
            poly T = C_reversed;
            C_reversed = adjust(C_reversed, B, d, b, m);
            L = n + 1 - L;
            B = T;
            b = d;
            m = 1;
        } else
            C_reversed = adjust(C_reversed, B, d, b, m++);
    }
    return std::make_pair(L, C_reversed);
}

// Calculate $x^N \bmod f(x)$
// Known as `Kitamasa method`
// Input: f_reversed: monic, reversed (f_reversed[0] = 1)
// Complexity: $O(K^2 \log N)$ ($K$: deg. of $f$)
// Example: (4, [1, -1, -1]) -> [2, 3]
//          ( x^4 = (x^2 + x + 2)(x^2 - x - 1) + 3x + 2 )
// Reference: http://misawa.github.io/others/fast_kitamasa_method.html
//            http://sugarknri.hatenablog.com/entry/2017/11/18/233936
template <typename Tfield>
std::vector<Tfield> monomial_mod_polynomial(long long N, const std::vector<Tfield> &f_reversed) {
    assert(!f_reversed.empty() and f_reversed[0] == 1);
    int K = f_reversed.size() - 1;
    if (!K) return {};
    int D = 64 - __builtin_clzll(N);
    std::vector<Tfield> ret(K, 0);
    ret[0] = 1;
    auto self_conv = [](std::vector<Tfield> x) -> std::vector<Tfield> {
        int d = x.size();
        std::vector<Tfield> ret(d * 2 - 1);
        for (int i = 0; i < d; i++) {
            ret[i * 2] += x[i] * x[i];
            for (int j = 0; j < i; j++) ret[i + j] += x[i] * x[j] * 2;
        }
        return ret;
    };
    for (int d = D; d--;) {
        ret = self_conv(ret);
        for (int i = 2 * K - 2; i >= K; i--) {
            for (int j = 1; j <= K; j++) ret[i - j] -= ret[i] * f_reversed[j];
        }
        ret.resize(K);
        if ((N >> d) & 1) {
            std::vector<Tfield> c(K);
            c[0] = -ret[K - 1] * f_reversed[K];
            for (int i = 1; i < K; i++) { c[i] = ret[i - 1] - ret[K - 1] * f_reversed[K - i]; }
            ret = c;
        }
    }
    return ret;
}

// Guess k-th element of the sequence, assuming linear recurrence
// initial_elements: 0-ORIGIN
// Verify: abc198f https://atcoder.jp/contests/abc198/submissions/21837815
template <typename Tfield>
Tfield guess_kth_term(const std::vector<Tfield> &initial_elements, long long k) {
    assert(k >= 0);
    if (k < static_cast<long long>(initial_elements.size())) return initial_elements[k];
    const auto f = find_linear_recurrence<Tfield>(initial_elements).second;
    const auto g = monomial_mod_polynomial<Tfield>(k, f);
    Tfield ret = 0;
    for (unsigned i = 0; i < g.size(); i++) ret += g[i] * initial_elements[i];
    return ret;
}
#line 3 "linear_algebra_matrix/blackbox_algorithm.hpp"
#include <chrono>
#include <random>
#line 6 "linear_algebra_matrix/blackbox_algorithm.hpp"

template <typename ModInt> std::vector<ModInt> gen_random_vector(int len) {
    static std::mt19937 mt(std::chrono::steady_clock::now().time_since_epoch().count());
    static std::uniform_int_distribution<int> rnd(1, ModInt::mod() - 1);
    std::vector<ModInt> ret(len);
    for (auto &x : ret) x = rnd(mt);
    return ret;
};

// Probabilistic algorithm to find a solution of linear equation Ax = b if exists.
// Complexity: O(n T(n) + n^2)
// Reference:
// [1] W. Eberly, E. Kaltofen, "On randomized Lanczos algorithms," Proc. of international symposium on
//     Symbolic and algebraic computation, 176-183, 1997.
template <typename Matrix, typename T>
std::vector<T> linear_system_solver_lanczos(const Matrix &A, const std::vector<T> &b) {
    assert(A.height() == int(b.size()));
    const int M = A.height(), N = A.width();

    const std::vector<T> D1 = gen_random_vector<T>(N), D2 = gen_random_vector<T>(M),
                         v = gen_random_vector<T>(N);
    auto applyD1 = [&D1](std::vector<T> v) {
        for (int i = 0; i < int(v.size()); i++) v[i] *= D1[i];
        return v;
    };
    auto applyD2 = [&D2](std::vector<T> v) {
        for (int i = 0; i < int(v.size()); i++) v[i] *= D2[i];
        return v;
    };
    auto applyAtilde = [&](std::vector<T> v) -> std::vector<T> {
        v = applyD1(v);
        v = A.prod(v);
        v = applyD2(v);
        v = A.prod_left(v);
        v = applyD1(v);
        return v;
    };
    auto dot = [&](const std::vector<T> &vl, const std::vector<T> &vr) -> T {
        return std::inner_product(vl.begin(), vl.end(), vr.begin(), T(0));
    };
    auto scalar_vec = [&](const T &x, std::vector<T> vec) -> std::vector<T> {
        for (auto &v : vec) v *= x;
        return vec;
    };

    auto btilde1 = applyD1(A.prod_left(applyD2(b))), btilde2 = applyAtilde(v);
    std::vector<T> btilde(N);
    for (int i = 0; i < N; i++) btilde[i] = btilde1[i] + btilde2[i];

    std::vector<T> w0 = btilde, v1 = applyAtilde(w0);
    std::vector<T> wm1(w0.size()), v0(v1.size());
    T t0 = dot(v1, w0), gamma = dot(btilde, w0) / t0, tm1 = 1;
    std::vector<T> x = scalar_vec(gamma, w0);
    while (true) {
        if (!t0 or !std::count_if(w0.begin(), w0.end(), [](T x) { return x != T(0); })) break;
        T alpha = dot(v1, v1) / t0, beta = dot(v1, v0) / tm1;
        std::vector<T> w1(N);
        for (int i = 0; i < N; i++) w1[i] = v1[i] - alpha * w0[i] - beta * wm1[i];
        std::vector<T> v2 = applyAtilde(w1);
        T t1 = dot(w1, v2);
        gamma = dot(btilde, w1) / t1;
        for (int i = 0; i < N; i++) x[i] += gamma * w1[i];

        wm1 = w0, w0 = w1;
        v0 = v1, v1 = v2;
        tm1 = t0, t0 = t1;
    }
    for (int i = 0; i < N; i++) x[i] -= v[i];
    return applyD1(x);
}

// Probabilistic algorithm to calculate determinant of matrices
// Complexity: O(n T(n) + n^2)
// Reference:
// [1] D. H. Wiedmann, "Solving sparse linear equations over finite fields,"
//     IEEE Trans. on Information Theory, 32(1), 54-62, 1986.
template <class Matrix, class Tp> Tp blackbox_determinant(const Matrix &M) {
    assert(M.height() == M.width());
    const int N = M.height();
    std::vector<Tp> b = gen_random_vector<Tp>(N), u = gen_random_vector<Tp>(N),
                    D = gen_random_vector<Tp>(N);
    std::vector<Tp> uMDib(2 * N);
    for (int i = 0; i < 2 * N; i++) {
        uMDib[i] = std::inner_product(u.begin(), u.end(), b.begin(), Tp(0));
        for (int j = 0; j < N; j++) b[j] *= D[j];
        b = M.prod(b);
    }
    auto ret = find_linear_recurrence<Tp>(uMDib);
    Tp det = ret.second.back() * (N % 2 ? -1 : 1);
    Tp ddet = 1;
    for (auto d : D) ddet *= d;
    return det / ddet;
}

// Complexity: O(n T(n) + n^2)
template <class Matrix, class Tp>
std::vector<Tp> reversed_minimal_polynomial_of_matrix(const Matrix &M) {
    assert(M.height() == M.width());
    const int N = M.height();
    std::vector<Tp> b = gen_random_vector<Tp>(N), u = gen_random_vector<Tp>(N);
    std::vector<Tp> uMb(2 * N);
    for (int i = 0; i < 2 * N; i++) {
        uMb[i] = std::inner_product(u.begin(), u.end(), b.begin(), Tp());
        b = M.prod(b);
    }
    auto ret = find_linear_recurrence<Tp>(uMb);
    return ret.second;
}

// Calculate A^k b
// Complexity: O(n^2 log k + n T(n))
// Verified: https://www.codechef.com/submit/COUNTSEQ2
template <class Matrix, class Tp>
std::vector<Tp> blackbox_matrix_pow_vec(const Matrix &A, long long k, std::vector<Tp> b) {
    assert(A.width() == int(b.size()));
    assert(k >= 0);

    std::vector<Tp> rev_min_poly = reversed_minimal_polynomial_of_matrix<Matrix, Tp>(A);
    std::vector<Tp> remainder = monomial_mod_polynomial<Tp>(k, rev_min_poly);

    std::vector<Tp> ret(b.size());
    for (Tp c : remainder) {
        for (int d = 0; d < int(b.size()); ++d) ret[d] += b[d] * c;
        b = A.prod(b);
    }
    return ret;
}
#line 3 "convolution/ntt.hpp"

#line 5 "convolution/ntt.hpp"
#include <array>
#line 7 "convolution/ntt.hpp"
#include <tuple>
#line 9 "convolution/ntt.hpp"

// CUT begin
// Integer convolution for arbitrary mod
// with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class.
// We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`.
// input: a (size: n), b (size: m)
// return: vector (size: n + m - 1)
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner);

constexpr int nttprimes[3] = {998244353, 167772161, 469762049};

// Integer FFT (Fast Fourier Transform) for ModInt class
// (Also known as Number Theoretic Transform, NTT)
// is_inverse: inverse transform
// ** Input size must be 2^n **
template <typename MODINT> void ntt(std::vector<MODINT> &a, bool is_inverse = false) {
    int n = a.size();
    if (n == 1) return;
    static const int mod = MODINT::mod();
    static const MODINT root = MODINT::get_primitive_root();
    assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0);

    static std::vector<MODINT> w{1}, iw{1};
    for (int m = w.size(); m < n / 2; m *= 2) {
        MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw;
        w.resize(m * 2), iw.resize(m * 2);
        for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv;
    }

    if (!is_inverse) {
        for (int m = n; m >>= 1;) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m] * w[k];
                    a[i] = x + y, a[i + m] = x - y;
                }
            }
        }
    } else {
        for (int m = 1; m < n; m *= 2) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m];
                    a[i] = x + y, a[i + m] = (x - y) * iw[k];
                }
            }
        }
        int n_inv = MODINT(n).inv().val();
        for (auto &v : a) v *= n_inv;
    }
}
template <int MOD>
std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) {
    int sz = a.size();
    assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
    std::vector<ModInt<MOD>> ap(sz), bp(sz);
    for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
    ntt(ap, false);
    if (a == b)
        bp = ap;
    else
        ntt(bp, false);
    for (int i = 0; i < sz; i++) ap[i] *= bp[i];
    ntt(ap, true);
    return ap;
}
long long garner_ntt_(int r0, int r1, int r2, int mod) {
    using mint2 = ModInt<nttprimes[2]>;
    static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
    static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv().val();
    static const long long m01_inv_m2 = mint2(m01).inv().val();

    int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
    auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2;
    return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val()) % mod;
}
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner) {
    if (a.empty() or b.empty()) return {};
    int sz = 1, n = a.size(), m = b.size();
    while (sz < n + m) sz <<= 1;
    if (sz <= 16) {
        std::vector<MODINT> ret(n + m - 1);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j];
        }
        return ret;
    }
    int mod = MODINT::mod();
    if (skip_garner or
        std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes)) {
        a.resize(sz), b.resize(sz);
        if (a == b) {
            ntt(a, false);
            b = a;
        } else {
            ntt(a, false), ntt(b, false);
        }
        for (int i = 0; i < sz; i++) a[i] *= b[i];
        ntt(a, true);
        a.resize(n + m - 1);
    } else {
        std::vector<int> ai(sz), bi(sz);
        for (int i = 0; i < n; i++) ai[i] = a[i].val();
        for (int i = 0; i < m; i++) bi[i] = b[i].val();
        auto ntt0 = nttconv_<nttprimes[0]>(ai, bi);
        auto ntt1 = nttconv_<nttprimes[1]>(ai, bi);
        auto ntt2 = nttconv_<nttprimes[2]>(ai, bi);
        a.resize(n + m - 1);
        for (int i = 0; i < n + m - 1; i++)
            a[i] = garner_ntt_(ntt0[i].val(), ntt1[i].val(), ntt2[i].val(), mod);
    }
    return a;
}

template <typename MODINT>
std::vector<MODINT> nttconv(const std::vector<MODINT> &a, const std::vector<MODINT> &b) {
    return nttconv<MODINT>(a, b, false);
}
#line 5 "linear_algebra_matrix/blackbox_matrices.hpp"
#include <numeric>
#line 8 "linear_algebra_matrix/blackbox_matrices.hpp"

// Sparse matrix
template <typename Tp> struct sparse_matrix {
    int H, W;
    std::vector<std::vector<std::pair<int, Tp>>> vals;
    sparse_matrix(int H = 0, int W = 0) : H(H), W(W), vals(H) {}
    int height() const { return H; }
    int width() const { return W; }
    void add_element(int i, int j, Tp val) {
        assert(i >= 0 and i < H);
        assert(j >= 0 and i < W);
        vals[i].emplace_back(j, val);
    }
    std::vector<Tp> prod(const std::vector<Tp> &vec) const {
        assert(W == int(vec.size()));
        std::vector<Tp> ret(H);
        for (int i = 0; i < H; i++) {
            for (const auto &p : vals[i]) ret[i] += p.second * vec[p.first];
        }
        return ret;
    }
    std::vector<Tp> prod_left(const std::vector<Tp> &vec) const {
        assert(H == int(vec.size()));
        std::vector<Tp> ret(W);
        for (int i = 0; i < H; i++) {
            for (const auto &p : vals[i]) ret[p.first] += p.second * vec[i];
        }
        return ret;
    }
    std::vector<std::vector<Tp>> vecvec() const {
        std::vector<std::vector<Tp>> ret(H, std::vector<Tp>(W));
        for (int i = 0; i < H; i++) {
            for (auto p : vals[i]) ret[i][p.first] += p.second;
        }
        return ret;
    }
};

// Toeplitz matrix
// M = [
// [ vw_0 vw_1 vw_2 ... ]
// [ vh_1 ...           ]
// [ vh_2 ...           ]
// [ ...                ] (vw_0 == vh_0)
// prod() / prod_left() : O((H + W) log (H + W))
template <typename NTTModInt> struct toeplitz_ntt {
    int _h, _w;
    int _len_pow2;
    std::vector<NTTModInt> ts, ntt_ts;
    toeplitz_ntt(const std::vector<NTTModInt> &vh, const std::vector<NTTModInt> &vw)
        : _h(vh.size()), _w(vw.size()) {
        assert(vh.size() and vw.size() and vh[0] == vw[0]);
        ts.resize(_h + _w - 1);
        for (int i = 0; i < _w; i++) ts[i] = vw[_w - 1 - i];
        for (int j = 0; j < _h; j++) ts[_w - 1 + j] = vh[j];
        _len_pow2 = 1;
        while (_len_pow2 < int(ts.size()) + std::max(_h, _w) - 1) _len_pow2 *= 2;
        ntt_ts = ts;
        ntt_ts.resize(_len_pow2);
        ntt(ntt_ts, false);
    }
    int height() const { return _h; }
    int width() const { return _w; }
    std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _w);
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _w - 1);
        b.resize(_h);
        return b;
    }
    std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _h);
        std::reverse(b.begin(), b.end());
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _h - 1);
        b.resize(_w);
        std::reverse(b.begin(), b.end());
        return b;
    }
    std::vector<std::vector<NTTModInt>> vecvec() const {
        std::vector<std::vector<NTTModInt>> ret(_h, std::vector<NTTModInt>(_w));
        for (int i = 0; i < _h; i++) {
            for (int j = 0; j < _w; j++) ret[i][j] = ts[i - j + _w - 1];
        }
        return ret;
    }
};

// Square Toeplitz matrix
// M = [
// [ t_(N-1) t_(N-2) ... t_1 t_0     ]
// [ t_N     t_(N-1) ... t_2 t_1     ]
// [ ...                             ]
// [ t_(N*2-2)       ...     t_(N-1) ]]
// prod() / prod_left() : O(N log N)
template <typename NTTModInt> struct square_toeplitz_ntt {
    int _dim;
    int _len_pow2;
    std::vector<NTTModInt> ts;
    std::vector<NTTModInt> ntt_ts;
    square_toeplitz_ntt(const std::vector<NTTModInt> &t) : _dim(t.size() / 2 + 1), ts(t) {
        assert(t.size() % 2);
        _len_pow2 = 1;
        while (_len_pow2 < int(ts.size()) + _dim - 1) _len_pow2 *= 2;
        ntt_ts = ts;
        ntt_ts.resize(_len_pow2);
        ntt(ntt_ts, false);
    }
    int height() const { return _dim; }
    int width() const { return _dim; }

    // Calculate A * b
    std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _dim);
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _dim - 1);
        b.resize(_dim);
        return b;
    }
    // Calculate bT * A
    std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const {
        std::reverse(b.begin(), b.end());
        b = prod(b);
        std::reverse(b.begin(), b.end());
        return b;
    }
    std::vector<std::vector<NTTModInt>> vecvec() const {
        std::vector<std::vector<NTTModInt>> ret(_dim, std::vector<NTTModInt>(_dim));
        for (int i = 0; i < _dim; i++) {
            for (int j = 0; j < _dim; j++) ret[i][j] = ts[i - j + _dim - 1];
        }
        return ret;
    }
};
#line 4 "linear_algebra_matrix/matrix.hpp"
#include <cmath>
#include <iterator>
#include <type_traits>
#line 9 "linear_algebra_matrix/matrix.hpp"

namespace matrix_ {
struct has_id_method_impl {
    template <class T_> static auto check(T_ *) -> decltype(T_::id(), std::true_type());
    template <class T_> static auto check(...) -> std::false_type;
};
template <class T_> struct has_id : decltype(has_id_method_impl::check<T_>(nullptr)) {};
} // namespace matrix_

template <typename T> struct matrix {
    int H, W;
    std::vector<T> elem;
    typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; }
    inline T &at(int i, int j) { return elem[i * W + j]; }
    inline T get(int i, int j) const { return elem[i * W + j]; }
    int height() const { return H; }
    int width() const { return W; }
    std::vector<std::vector<T>> vecvec() const {
        std::vector<std::vector<T>> ret(H);
        for (int i = 0; i < H; i++) {
            std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i]));
        }
        return ret;
    }
    operator std::vector<std::vector<T>>() const { return vecvec(); }
    matrix() = default;
    matrix(int H, int W) : H(H), W(W), elem(H * W) {}
    matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) {
        for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem));
    }

    template <typename T2, typename std::enable_if<matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2::id();
    }
    template <typename T2, typename std::enable_if<!matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2(1);
    }

    static matrix Identity(int N) {
        matrix ret(N, N);
        for (int i = 0; i < N; i++) ret.at(i, i) = _T_id<T>();
        return ret;
    }

    matrix operator-() const {
        matrix ret(H, W);
        for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i];
        return ret;
    }
    matrix operator*(const T &v) const {
        matrix ret = *this;
        for (auto &x : ret.elem) x *= v;
        return ret;
    }
    matrix operator/(const T &v) const {
        matrix ret = *this;
        const T vinv = _T_id<T>() / v;
        for (auto &x : ret.elem) x *= vinv;
        return ret;
    }
    matrix operator+(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i];
        return ret;
    }
    matrix operator-(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i];
        return ret;
    }
    matrix operator*(const matrix &r) const {
        matrix ret(H, r.W);
        for (int i = 0; i < H; i++) {
            for (int k = 0; k < W; k++) {
                for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j);
            }
        }
        return ret;
    }
    matrix &operator*=(const T &v) { return *this = *this * v; }
    matrix &operator/=(const T &v) { return *this = *this / v; }
    matrix &operator+=(const matrix &r) { return *this = *this + r; }
    matrix &operator-=(const matrix &r) { return *this = *this - r; }
    matrix &operator*=(const matrix &r) { return *this = *this * r; }
    bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; }
    bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; }
    bool operator<(const matrix &r) const { return elem < r.elem; }
    matrix pow(int64_t n) const {
        matrix ret = Identity(H);
        bool ret_is_id = true;
        if (n == 0) return ret;
        for (int i = 63 - __builtin_clzll(n); i >= 0; i--) {
            if (!ret_is_id) ret *= ret;
            if ((n >> i) & 1) ret *= (*this), ret_is_id = false;
        }
        return ret;
    }
    std::vector<T> pow_vec(int64_t n, std::vector<T> vec) const {
        matrix x = *this;
        while (n) {
            if (n & 1) vec = x * vec;
            x *= x;
            n >>= 1;
        }
        return vec;
    };
    matrix transpose() const {
        matrix ret(W, H);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j);
        }
        return ret;
    }
    // Gauss-Jordan elimination
    // - Require inverse for every non-zero element
    // - Complexity: O(H^2 W)
    template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        int piv = -1;
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c))))
                piv = j;
        }
        return piv;
    }
    template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) != T2()) return j;
        }
        return -1;
    }
    matrix gauss_jordan() const {
        int c = 0;
        matrix mtr(*this);
        std::vector<int> ws;
        ws.reserve(W);
        for (int h = 0; h < H; h++) {
            if (c == W) break;
            int piv = choose_pivot(mtr, h, c);
            if (piv == -1) {
                c++;
                h--;
                continue;
            }
            if (h != piv) {
                for (int w = 0; w < W; w++) {
                    std::swap(mtr[piv][w], mtr[h][w]);
                    mtr.at(piv, w) *= -_T_id<T>(); // To preserve sign of determinant
                }
            }
            ws.clear();
            for (int w = c; w < W; w++) {
                if (mtr.at(h, w) != T()) ws.emplace_back(w);
            }
            const T hcinv = _T_id<T>() / mtr.at(h, c);
            for (int hh = 0; hh < H; hh++)
                if (hh != h) {
                    const T coeff = mtr.at(hh, c) * hcinv;
                    for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff;
                    mtr.at(hh, c) = T();
                }
            c++;
        }
        return mtr;
    }
    int rank_of_gauss_jordan() const {
        for (int i = H * W - 1; i >= 0; i--) {
            if (elem[i] != 0) return i / W + 1;
        }
        return 0;
    }
    int rank() const { return gauss_jordan().rank_of_gauss_jordan(); }

    T determinant_of_upper_triangle() const {
        T ret = _T_id<T>();
        for (int i = 0; i < H; i++) ret *= get(i, i);
        return ret;
    }
    int inverse() {
        assert(H == W);
        std::vector<std::vector<T>> ret = Identity(H), tmp = *this;
        int rank = 0;
        for (int i = 0; i < H; i++) {
            int ti = i;
            while (ti < H and tmp[ti][i] == T()) ti++;
            if (ti == H) {
                continue;
            } else {
                rank++;
            }
            ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]);
            T inv = _T_id<T>() / tmp[i][i];
            for (int j = 0; j < W; j++) ret[i][j] *= inv;
            for (int j = i + 1; j < W; j++) tmp[i][j] *= inv;
            for (int h = 0; h < H; h++) {
                if (i == h) continue;
                const T c = -tmp[h][i];
                for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c;
                for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c;
            }
        }
        *this = ret;
        return rank;
    }
    friend std::vector<T> operator*(const matrix &m, const std::vector<T> &v) {
        assert(m.W == int(v.size()));
        std::vector<T> ret(m.H);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j];
        }
        return ret;
    }
    friend std::vector<T> operator*(const std::vector<T> &v, const matrix &m) {
        assert(int(v.size()) == m.H);
        std::vector<T> ret(m.W);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j);
        }
        return ret;
    }
    std::vector<T> prod(const std::vector<T> &v) const { return (*this) * v; }
    std::vector<T> prod_left(const std::vector<T> &v) const { return v * (*this); }
    template <class OStream> friend OStream &operator<<(OStream &os, const matrix &x) {
        os << "[(" << x.H << " * " << x.W << " matrix)";
        os << "\n[column sums: ";
        for (int j = 0; j < x.W; j++) {
            T s = T();
            for (int i = 0; i < x.H; i++) s += x.get(i, j);
            os << s << ",";
        }
        os << "]";
        for (int i = 0; i < x.H; i++) {
            os << "\n[";
            for (int j = 0; j < x.W; j++) os << x.get(i, j) << ",";
            os << "]";
        }
        os << "]\n";
        return os;
    }
    template <class IStream> friend IStream &operator>>(IStream &is, matrix &x) {
        for (auto &v : x.elem) is >> v;
        return is;
    }
};
#line 7 "linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp"
#include <cstdio>
#line 11 "linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp"
using namespace std;

mt19937 mt(1010101);

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, toeplitz_ntt<MODINT>>::value>::type * = nullptr>
toeplitz_ntt<MODINT> gen_random_matrix() {
    auto rnd = uniform_int_distribution<int>(1, 100);
    int h = rnd(mt), w = max(h, rnd(mt));
    auto vh = gen_random_vector<MODINT>(h), vw = gen_random_vector<MODINT>(w);
    vw[0] = vh[0];
    return toeplitz_ntt<MODINT>(vh, vw);
}

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, square_toeplitz_ntt<MODINT>>::value>::type * = nullptr>
square_toeplitz_ntt<MODINT> gen_random_matrix() {
    int N = uniform_int_distribution<int>(1, 100)(mt);
    auto v = gen_random_vector<MODINT>(N * 2 - 1);
    return square_toeplitz_ntt<MODINT>(v);
}

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, sparse_matrix<MODINT>>::value>::type * = nullptr>
sparse_matrix<MODINT> gen_random_matrix() {
    int H = uniform_int_distribution<int>(1, 20)(mt),
        W = max(H, uniform_int_distribution<int>(1, 20)(mt));
    sparse_matrix<MODINT> M(H, W);
    for (int i = 0; i < H; i++) {
        for (int j = 0; j < W; j++) {
            MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt);
            M.add_element(i, j, v);
        }
    }
    return M;
}

template <class MODINT, class MATRIX,
          typename std::enable_if<std::is_same<MATRIX, matrix<MODINT>>::value>::type * = nullptr>
matrix<MODINT> gen_random_matrix() {
    int H = uniform_int_distribution<int>(1, 20)(mt),
        W = max(H, uniform_int_distribution<int>(1, 20)(mt));
    matrix<MODINT> M(H, W);
    for (int i = 0; i < H; i++) {
        for (int j = 0; j < W; j++) {
            MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt);
            M[i][j] = v;
        }
    }
    return M;
}

template <typename MATRIX, typename mint> void blackbox_mat_checker(int n_try) {
    for (int i = 0; i < n_try; i++) {
        const MATRIX M = gen_random_matrix<mint, MATRIX>();
        vector<vector<mint>> Mvv = M.vecvec();
        const int H = M.height(), W = M.width();
        const auto b = gen_random_vector<mint>(H);
        // Check linear eq. solver
        const auto x = linear_system_solver_lanczos(M, b);
        assert(M.prod(x) == b);

        // Check prod()
        const auto c = gen_random_vector<mint>(W);
        const auto Mc = M.prod(c);
        vector<mint> Mc2(H);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) Mc2[i] += Mvv[i][j] * c[j];
        }
        assert(Mc == Mc2);

        // Check prod_left()
        const auto a = gen_random_vector<mint>(H);
        const auto aM = M.prod_left(a);
        vector<mint> aM2(W);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) aM2[j] += Mvv[i][j] * a[i];
        }
        assert(aM == aM2);

        // Check black_box_determinant()
        if (H == W) {
            mint det = blackbox_determinant<MATRIX, mint>(M);
            mint det2 = matrix<mint>(Mvv).gauss_jordan().determinant_of_upper_triangle();
            assert(det == det2);
        }
    }
}

int main() {
    using mint = ModInt<998244353>;
    blackbox_mat_checker<toeplitz_ntt<mint>, mint>(1000);
    blackbox_mat_checker<square_toeplitz_ntt<mint>, mint>(1000);
    blackbox_mat_checker<sparse_matrix<mint>, mint>(1000);
    blackbox_mat_checker<matrix<mint>, mint>(1000);
    puts("Hello World");
}

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